So, something like “fraction of preferred states shared” ?
Describe preferred states for P1 as cells in the payoff matrix that are best for P1 for each P2 action (and preferred stated for P2 in a similar manner)
Fraction of P1 preferred states that are also preferred for P2 is measurement of alignment P1 to P2.
Fraction of shared states between players to total number of preferred states is measure of total alignment of the game.
For 2x2 game each player will have 2 preferred states (corresponding to the 2 possible action of the opponent). If 1 of them will be the same cell that will mean that each player is 50% aligned to other (1 of 2 shared) and the game in total is 33% aligned (1 of 3), This also generalize easily to NxN case and for >2 players.
And if there are K multiple cells with the same payoff to choose from for some opponent action we can give 1/K to them instead of 1.
(it would be much easier to explain with a picture and/or table, but I’m pretty new here and wasn’t able to find how to do them here yet)
Does agency matter? There are 21 x 21 x 4 possible payoff matrixes for a 2x2 game if we use Ordinal payoffs. For the vast majority of them (all but about 7 x 7 x 4 of them) , one or both players can make a decision without knowing or caring what the other player’s payoffs are, and get the best possible result. Of the remaining 182 arrangements, 55 have exactly one box where both players get their #1 payoff (and, therefore, will easily select that as the equilibrium).
All the interesting choices happen in the other 128ish arrangements, 6⁄7 of which have the pattern of the preferred (1st and 1st, or 1st and 2nd) options being on a diagonal. The most interesting one (for the player picking the row, and getting the first payoff) is:
1 / (2, 3, or 4) ; 4 / (any)
2 / (any) ; 3 / (any)
The optimal strategy for any interesting layout will be a mixed strategy, with the % split dependent on the relative Cardinal payoffs (which are generally not calculatable since they include Reputation and other non-quantifiable effects).
Therefore, you would want to weight the quality of any particular result by the chance of that result being achieved (which also works for the degenerate cases where one box gets 100% of the results, or two perfectly equivalent boxes share that)
I like this answer, and I’m going to take more time to chew on it.
I have put the preferred state for each player in bold. I think by your rule this works out to 50% aligned. However, the Nash equilibrium is both players choosing the 1⁄1 result, which seems perfectly aligned (intuitively).
In this game, all preferred states are shared, yet there is a Nash equilibrium where each player plays the move that can get them 1 point 2⁄3 of the time, and the other move 1⁄3 of the time. I think it would be incorrect to call this 100% aligned.
(These examples were not obvious to me, and tracking them down helped me appreciate the question more. Thank you.)
Thanks for careful analysis, I must confess that my metric does not consider the stochastic strategies, and in general works better if players actions are taken consequently, not simultaneously (which is much different from the classic description).
The reasoning being that for maximal alignment each action of P1 there exist exactly one action of P2 (and vice versa) that is Nash equilibrium. In this case the game stops in stable state after single pair of actions. And maximally unaligned game will have no nash equilibrium at all, meaning the players actions-reactions will just move over the matrix in closed loop.
Overall, my solution as is seems not fitted for the classical formulation of the game :) but thanks for considering it!